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March, 1993 Differentiability of Statistical Functionals and Consistency of the Jackknife
Jun Shao
Ann. Statist. 21(1): 61-75 (March, 1993). DOI: 10.1214/aos/1176349015

Abstract

In statistical applications the unknown parameter of interest can frequently be defined as a functional $\theta=T(F)$, where F is an unknown population. Statistical inferences about $\theta$ are usually made based on the statistic $T(F_n)$, where $F_n$ is the empirical distribution. Assessing $T(F_n)$ (as an estimator of $\theta$) or making large sample inferences usually requires a consistent estimator of the asymptotic variance of $T(F_n)$. Asymptotic behavior of the jackknife variance estimator is closely related to the smoothness of the functional T. This paper studies the smoothness of T through the differentiability of T and establishes some general results for the consistency of the jackknife variance estimators. The results are applied to some examples in which the statistics $T(F_n)$ are L-, M-estimators and some test statistics.

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Jun Shao. "Differentiability of Statistical Functionals and Consistency of the Jackknife." Ann. Statist. 21 (1) 61 - 75, March, 1993. https://doi.org/10.1214/aos/1176349015

Information

Published: March, 1993
First available in Project Euclid: 12 April 2007

zbMATH: 0783.62028
MathSciNet: MR1212166
Digital Object Identifier: 10.1214/aos/1176349015

Subjects:
Primary: 62G05
Secondary: 62E99

Keywords: Continuous differentiability , Frechet differentiability , Gateaux derivative , L-estimators , Linear rank statistics , M-estimators , variance estimation

Rights: Copyright © 1993 Institute of Mathematical Statistics

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